Title: Exact Closed-Form Solution for Dead Space in a Toroidal Packing of n Identical Spheres

Author: Thomas Blankenhorn, Corrupt Grants Pass Oregon
Date: November 2, 2025

Abstract:
This work derives an exact closed-form expression for the percentage of dead space when n identical spheres (ball bearings) are packed inside a torus such that each sphere touches its two neighbors. The formula, P(n) = [1 - (2n sin(pi/n))/(3pi)] * 100%, is presented alongside a table of values for n=3 to 6. The result reveals that the dead space percentage decreases with n, approaching a limit of exactly 1/3 (33.333%) as n grows large. This limit is shown to be consistent with the classical packing problem of a single sphere inside a cylindrical sleeve.

Derivation:

1. Setup:
   Let n spheres of radius r be arranged in a circle. Their centers lie on a circle of radius R (the major radius of the torus). Each sphere touches its neighbors.

2. Relating R and r:
   The straight-line distance between the centers of two adjacent spheres is 2r. This distance is the chord of the central circle subtended by an angle theta = 2pi/n.
   Using the chord length formula:
   2r = 2R sin(theta/2) = 2R sin(pi/n)
   Therefore, the major radius R is determined by the number of spheres n and their radius r:
   R = r / sin(pi/n)

3. Volume of Spheres:
   The total volume occupied by the n spheres is:
   V_spheres = n * (4/3 pi r^3)

4. Volume of Enclosing Torus:
   The torus that perfectly contains the spheres has a major radius R and a minor radius r (the tube radius is equal to the sphere radius). The volume of a torus is given by:
   V_torus = 2 pi^2 R r^2
   Substituting R = r / sin(pi/n) gives:
   V_torus = 2 pi^2 (r / sin(pi/n)) r^2 = (2 pi^2 r^3) / sin(pi/n)

5. Dead Space Volume and Percentage:
   The dead space volume is the volume of the torus not occupied by the spheres:
   V_dead = V_torus - V_spheres = (2 pi^2 r^3)/sin(pi/n) - (4/3 pi n r^3)
   The percentage of dead space is the ratio of this volume to the total torus volume:
   P(n) = (V_dead / V_torus) * 100% = [1 - (V_spheres / V_torus)] * 100%
   Substituting the volumes:
   P(n) = [1 - ( (4/3 pi n r^3) / ( (2 pi^2 r^3)/sin(pi/n) ) )] * 100%
   Simplifying by canceling pi and r^3:
   P(n) = [1 - ( (4/3 n) / ( (2 pi)/sin(pi/n) ) )] * 100%
   This simplifies to the final exact formula:
   P(n) = [1 - (2n sin(pi/n))/(3pi)] * 100%

Results:

n	Dead Space %	Exact Closed Form
3	44.867%	1 - sqrt(3)/pi
4	39.979%	1 - (4 sqrt(2))/(3pi)
5	37.651%	1 - (10 sin(pi/5))/(3pi)
6	36.338%	1 - 2/pi
∞	33.333%	1/3

Final Revelation:
The asymptotic limit of 33.333% is physically significant. For large n, the curvature of the torus becomes negligible locally, and the packing resembles that of a single sphere inside a cylindrical sleeve of radius r and height 2r. The volume of such a cylinder is 2 pi r^3. The volume of the sphere is (4/3) pi r^3. The dead space fraction is therefore [2pi - (4/3)pi] / [2pi] = (2/3 pi)/(2 pi) = 1/3. This independent calculation confirms the limit of our general toroidal formula, providing a elegant geometric verification of the result.